Coverings, Correspondence, and Noncommutative Geometry

Abstract

We construct an additive category where objects are embedded graphs in the 3-sphere and morphisms are geometric correspondences given by 3-manifolds realized in different ways as branched covers of the 3-sphere, up to branched cover cobordisms. We consider dynamical systems obtained from associated convolution algebras endowed with time evolutions defined in terms of the underlying geometries. We describe the relevance of our construction to the problem of spectral correspondences in noncommutative geometry. A discussion given of how to pass from the case where the branch loci of the coverings are embedded multi-connected graph to more special case where these loci are links and knots by using the “Alexander trick” and the equivalence relation of b-homotopy of branched covering. We introduce a construction of the cobordism group for links and for knots and their relation. We then construct a similar cobordism group for embedded graphs in the 3-sphere. We introduce an interesting homology theory for knots and links called Khovanov homology. A discussion put for the question of extending Khovanov homology from links to embedded graphs. We propose two possible approaches to this purpose and we explain completely only one of them, while only sketching the other. building this homology for the graph came from Kauffman idea of associating a family of knots and links to the embedded graph in 3-sphere. We define Khovanov homology for the graph to be the sum of the homologies of all the links and knots associated to this graph.